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Orbifold partition function and their phases

We discuss a partition function of 3d supersymmetric gauge
theories on the (p, -1) Lens space.

In 3d the partition function is directly used to check dualities though the normalization is not seriously treated, especially, the phase is usually
ignored. However, when we consider the partition function on the orbifold the partition function consists of the sum of factors labeled by holonomies

and their relative
phases become crucial. We stress that the known formula for the partition function is incorrect and some relative phase factors are needed to identify the partition functions of
dual theories.